Guarantees by Construction for Learned Finite Volume Schemes on Steady Supersonic Flow

What the math gives to ML

The paper replaces penalty-based physical admissibility with a parameterization whose outputs are safe for every neural-network weight configuration. Its transferable asset is the separation between a learned local reconstruction and a deterministic invariant-domain wrapper: the network can improve accuracy, while density and pressure positivity are enforced algebraically rather than learned statistically. The same pattern can be used for neural finite-volume solvers, differentiable simulators, and graph message-passing operators that must preserve bounded or physically admissible states. The most practical experiment is to learn local reconstructions while computing the limiter by explicit projection onto the admissible state set.